Acta mathematica scientia, Series B >
ON SOLUTIONS OF QUATERNION MATRIX EQUATIONS XF -AX = BY AND XF-AX = BY
Received date: 2010-12-31
Revised date: 2012-02-20
Online published: 2012-09-20
Supported by
This project is granted financial support from NSFC (11071079), NSFC (10901056), Shanghai Science and Technology Commission Venus (11QA1402200), Ningbo Natural Science Foundation (2010A610097), the Fundamental Research Funds for the Central Universi-
ties and Zhejiang Natural Science Foundation (Y6110043).
In this paper, the quaternion matrix equations XF − AX = BY and XF −AX= BY are investigated. For convenience, they were called generalized Sylvester-quaternion matrix equation and generalized Sylvester-j-conjugate quaternion matrix equa-tion, which include the Sylvester matrix equation and Lyapunov matrix equation as special cases. By applying of Kronecker map and complex representation of a quaternion matrix, the sufficient conditions to compute the solution can be given and the expressions of the explicit solutions to the above two quaternion matrix equations XF − AX = BY and XF − AX = BY are also obtained. By the established expressions, it is easy to compute
the solution of the quaternion matrix equation in the above two forms. In addition, two practical algorithms for these two quaternion matrix equations are give. One is complex representation matrix method and the other is a direct algorithm by the given expression. Furthermore, two illustrative examples are proposed to show the efficiency of the given method.
SONG Cai-Qin , CHEN Guan-Liang , WANG Xiao-Dong . ON SOLUTIONS OF QUATERNION MATRIX EQUATIONS XF -AX = BY AND XF-AX = BY[J]. Acta mathematica scientia, Series B, 2012 , 32(5) : 1967 -1982 . DOI: 10.1016/S0252-9602(12)60153-2
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